Molecular Integrals
In quantum chemistry, the very well known Hartree-Fock (HF) method is used to approximate the ground-state electronic structure of a molecule. The method involves solving the HF equation, which is given by $$ \hat{f} | \varphi_i \rangle = \epsilon_i | \varphi_i \rangle $$ where \(\hat{f}\) is the Fock operator, \(| \varphi_i \rangle \) is the \(i\)-th eigenfunction, \(\epsilon_i\) is the corresponding eigenvalue. We identify the eigenfunction with canonical molecular orbitals and the eigenvalues with orbital energies.
The Fock operator is defined as $$ \hat{f} = \hat{h}^{\mathrm{core}} + \sum_{j=1}^{N} \left( \hat{J}_j - \hat{K}_j \right) $$ with the one-electron Hamiltonian \(\hat{h}^{\mathrm{core}}\), the Coulomb operator \(\hat{J}_j\) and the exchange operator \(\hat{K}_j\), and \(N\) stands for the number of electrons.
Since \(\hat{h}^{\mathrm{core}}\) contains differential operators and \(\hat{J}_j\) and \(\hat{K}_j\) have integrals in them, the HF equation is an integro-differential equation, for which a closed-form solution is extremly difficult to obtain and analytical solutions are only known for the simplest cases.
Therefore, we utilize numerical methods to solve the HF equation. Though popular in other fields, a discretization is very impractical hier. Considering the rapid change of electronic density from nuclei to bonds and bond lengths of roughly 1 Å, at least 4 points per Å should be used for a crude representation of the wavefunction. Also, we should add a boundary to properly describe the fall-off of the wavefunction. For a medium-sized molecule, e.g. porphin, which is around 10 Å across, a box of the dimension 15 Å × 15 Å × 5 Å would be appropriate, which translates to 60 × 60 × 20 = 72000 grid points. This is far from practical. If we wish a finer granulated grid, or calculations for larger molecules, discretization will become infeasible rather quickly.
So, the spatial grid is a very inefficient basis for the HF equation. Because the molecule consists of atoms, it should be possible to represent the molecular orbitals with some sort of combinations of atomic orbitals. The simplest combination is the linear combination. In this case, our basis are atom-centered wavefunctions \(\chi_\mu\), and the molecular orbitals can be expressed as $$ | \varphi_i \rangle = \sum_{\mu} c_{\mu i} \chi_\mu $$ where \(\mu\) indexes the atomic basis functions.
Inserting this Ansatz into the HF equation and projecting both sides onto \(\langle \chi_\nu|\), we obtain $$ \begin{align} \langle \chi_\nu | \hat{f} | \sum_{\mu} c_{\mu i} \chi_\mu \rangle &= \epsilon_i \langle \chi_\nu | \sum_{\mu} c_{\mu i} \chi_\mu \rangle \\ \sum_{\mu} \langle \chi_\nu | \hat{f} | \chi_\mu \rangle c_{\mu i} &= \sum_{\mu} \langle \chi_\nu | \chi_\mu \rangle c_{\mu i} \epsilon_i \\ \mathbf{F}\vec{c}_i &= \mathbf{S}\vec{c}_i\epsilon_i \end{align} $$ where \(\mathbf{F}\) is the Fock matrix, \(\vec{c}_i\) is the coefficient vector of the \(i\)-th molecular orbital, \(\mathbf{S}\) is the overlap matrix, and \(\epsilon_i\) is the energy of the \(i\)-th molecular orbital.
The new equation is called Roothaan-Hall equation, where the difficult derivatives and integrals of the unknown molecular orbitals is reduced to derivatives and integrals of known basis functions. After evaluating these integrals in \(\mathbf{F}\) and \(\mathbf{S}\), the actual solving step is easily done using some linear algebra.
A closer inspection of the matrices \(\mathbf{F}\) and \(\mathbf{S}\) reveals that four types of molecular integrals exist:
- Overlap integrals: \(\langle \chi_\mu | \chi_\nu \rangle\)
- Kinetic energy integrals: \(\langle \chi_\mu | -\frac{1}{2} \nabla^2 | \chi_\nu \rangle\)
- nuclear attraction integrals: \(\langle \chi_\mu | -Z_{\mathrm{nuc}}/R_{\mathrm{nuc}} | \chi_\nu \rangle\)
- electron repulsion integrals: \(\langle \mu \nu | \lambda \sigma \rangle\)
In this chapter, some basis concepts of basis functions will be introduced, followed by symbolic calculation of closed-form expressions for molecular integrals. In the end, we will use these expressions to generate a module, which performs all these integrals.